Eating cheap is easy. It's just not convenient, that's why people think food is necessarily expensive, they are lazy or just stupid. Unless you are talking keeping it under 50 cents a day per person, you don't really even need to specifically look for the cheapest foods, you just need to buy unprepared foods, then cook it yourself. That's 90% of the work if you're trying to save money.
Think of the better/premium/fancier versions of certain cheap, classic staple food items like pinto beans and white rice.
Even if you went with quinoa and black beans, its still really cheap if you buy in bulk, especially when you consider how far you can stretch each pound of these.
And do people not even notice the fact that grocery stores are basically giving fresh vegetables away for free? Unless you insist on shopping at Whole Foods, a $10 bill will buy you a truckload of the stuff. Also eggs, maybe the best/healthiest food per $ on the planet.
Here's what I cooked last week, and it's my goto for workplace potlucks and things like that:
8oz bacon
1lb chorizo
1.5 lb dry black beans
1.5 lb dry small red kidney beans
~4 jalapeno's
~6-8 serrano's
1/2 a red onion
Whatever spices I feel like.
Bring beans to a boil and let them sit. Drain them, mostly. Chop up all the other shit and throw it all together in a crock pot over night. You're done. You just made a shitload of food for a few bucks.
Some things I'm always on the lookout for, even if I don't feel like eating it right now, since they keep forever in freezer or pantry:
different beans, lentils, chickpeas, that sort of thing
rice/grains
whole turkeys (local grocery store had whole turkey breasts on sale lately for 0.49/lb lately)
frozen mixed vegetables
+1 to people mentioning Aldi. If you're picky(and you shouldn't be if you're not a child or an asshole) they don't always have the exact thing you went in for since their inventory rotates, but plenty of good deals for a smart shopper.